2019 AIME I Problems/Problem 7
There are positive integers and that satisfy the system of equations Let be the number of (not necessarily distinct) prime factors in the prime factorization of , and let be the number of (not necessarily distinct) prime factors in the prime factorization of . Find .
Add the two equations to get that . Then, we use the theorem to get the equation, . Using the theorem that , along with the previously mentioned theorem, we can get the equation . This can easily be simplified to , or .
can be factored into , and equals to the sum of the exponents of and , which is . Multiply by two to get , which is . Then, use the first equation () to show that has to have lower degrees of and than (you can also test when , which is a contradiction to the restrains we set before). Therefore, . Then, turn the equation into , which yields , or . Factor this into , and add the two 20's, resulting in , which is . Add to (which is ) to get .
~minor mistake fix by virjoy2001 ~minor mistake fix by oralayhan
Remark: You can obtain the contradiction by using LTE. If . However, a contradiction. Same goes with taking
First simplifying the first and second equations, we get that
Thus, when the two equations are added, we have that When simplified, this equals so this means that so
Now, the following cannot be done on a proof contest but let's (intuitively) assume that and and are both powers of . This means the first equation would simplify to and Therefore, and and if we plug these values back, it works! has total factors and has so
Please remember that you should only assume on these math contests because they are timed; this would technically not be a valid solution.
Solution 3 (Easy Solution)
Let and and . Then the given equations become and . Therefore, and . Our answer is .
We will use the notation for and as . We can start with a similar way to Solution 1. We have, by logarithm properties, or . We can do something similar to the second equation and our two equations become Adding the two equations gives us . Since we know that , , or . We can express as and as . Another way to express is now , and is now . We know that , and thus, , and . Our equations for and now become or . Doing the same for the equation, we have , and , which satisfies . Thus, . ~awsomek
Let . Simplifying, . Notice that since are coprime, and (Prove it yourself !) , . Hence, giving the answer .
(Solution by Prabh1512)
Solution 6 (Official MAA)
The two equations are equivalent to and respectively. Multiplying corresponding sides of the equations leads to , so . It follows that there are nonnegative integers and such that with . Furthermore, Thus Because neither nor can equal when it follows that . Hence , so the prime factorization of has prime factors, and the prime factorization of has prime factors. The requested sum is
Add the two equations and use the fact that to find that . So let and for . If then the exponent of in is , so , contradiction. So . Then the exponent of in is , so . Similarly, . Then as desired.
~from trumpeter in the AoPS Forums Contest Discussion
Video Solution(Pretty Straightforward)
https://www.youtube.com/watch?v=NOLk9-A4eDo Remember to subscribe!
~North America math Contest Go Go Go
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